1 1 Slide Chapter 10 Project Scheduling: PERT/CPM n Project Scheduling with Known Activity Times n Project Scheduling with Uncertain Activity Times n Considering

1 1 Slide Slide

Chapter 10Chapter 10Project Scheduling: PERT/CPMProject Scheduling: PERT/CPM

Project Scheduling with Known Activity TimesProject Scheduling with Known Activity Times Project Scheduling with Uncertain Activity Project Scheduling with Uncertain Activity

TimesTimes Considering Time-Cost Trade-OffsConsidering Time-Cost Trade-Offs

2 2 Slide Slide

PERT/CPMPERT/CPM

PERTPERT• Program Evaluation and Review TechniqueProgram Evaluation and Review Technique

•Developed by U.S. Navy for Polaris missile Developed by U.S. Navy for Polaris missile projectproject

•Developed to handle uncertain activity timesDeveloped to handle uncertain activity times CPMCPM

•Critical Path MethodCritical Path Method

•Developed by Du Pont & Remington RandDeveloped by Du Pont & Remington Rand

•Developed for industrial projects for which Developed for industrial projects for which activity times generally were knownactivity times generally were known

Today’s project management software packages Today’s project management software packages have combined the best features of both have combined the best features of both approaches.approaches.

3 3 Slide Slide

PERT/CPMPERT/CPM

PERT and CPM have been used to plan, schedule, PERT and CPM have been used to plan, schedule, and control a wide variety of projects:and control a wide variety of projects:

•R&D of new products and processesR&D of new products and processes

•Construction of buildings and highwaysConstruction of buildings and highways

•Maintenance of large and complex equipmentMaintenance of large and complex equipment

•Design and installation of new systemsDesign and installation of new systems

4 4 Slide Slide

PERT/CPMPERT/CPM

PERT/CPM is used to plan the scheduling of PERT/CPM is used to plan the scheduling of individual individual activitiesactivities that make up a project. that make up a project.

Projects may have as many as several thousand Projects may have as many as several thousand activities.activities.

A complicating factor in carrying out the A complicating factor in carrying out the activities is that some activities depend on the activities is that some activities depend on the completion of other activities before they can be completion of other activities before they can be started.started.

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PERT/CPMPERT/CPM

Project managers rely on PERT/CPM to help them Project managers rely on PERT/CPM to help them answer questions such as:answer questions such as:

•What is the What is the total timetotal time to complete the project? to complete the project?

•What are the What are the scheduled start and finish datesscheduled start and finish dates for each specific activity?for each specific activity?

•Which activities are Which activities are criticalcritical and must be and must be completed exactly as scheduled to keep the completed exactly as scheduled to keep the project on schedule?project on schedule?

•How long can How long can noncritical activitiesnoncritical activities be delayed be delayed before they cause an increase in the project before they cause an increase in the project completion time?completion time?

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Project NetworkProject Network

A A project networkproject network can be constructed to model can be constructed to model the precedence of the activities. the precedence of the activities.

The The nodesnodes of the network represent the of the network represent the activities. activities.

The The arcsarcs of the network reflect the precedence of the network reflect the precedence relationships of the activities. relationships of the activities.

A A critical pathcritical path for the network is a path for the network is a path consisting of activities with zero slack.consisting of activities with zero slack.

7 7 Slide Slide

Example: Frank’s Fine FloatsExample: Frank’s Fine Floats

Frank’s Fine Floats is in the business of Frank’s Fine Floats is in the business of building elaborate parade floats. Frank and his building elaborate parade floats. Frank and his crew have a new float to build and want to use crew have a new float to build and want to use PERT/CPM to help them manage the projectPERT/CPM to help them manage the project . .

The table on the next slide shows the The table on the next slide shows the activities that comprise the project. Each activities that comprise the project. Each activity’s estimated completion time (in days) activity’s estimated completion time (in days) and immediate predecessors are listed as well.and immediate predecessors are listed as well.

Frank wants to know the total time to Frank wants to know the total time to complete the project, which activities are critical, complete the project, which activities are critical, and the earliest and latest start and finish dates and the earliest and latest start and finish dates for each activity.for each activity.

8 8 Slide Slide


Immediate CompletionImmediate Completion

ActivityActivity DescriptionDescription PredecessorsPredecessors Time (days)Time (days)

A Initial Paperwork A Initial Paperwork --- --- 3 3

B Build Body B Build Body A A 3 3

C Build Frame C Build Frame A A 2 2

D Finish Body D Finish Body B B 3 3

E Finish Frame E Finish Frame C C 7 7

F Final Paperwork F Final Paperwork B,C B,C 3 3

G Mount Body to Frame D,EG Mount Body to Frame D,E 6 6

H Install Skirt on Frame CH Install Skirt on Frame C 2 2

9 9 Slide Slide



StartStart FinishFinish

BB33

DD33

AA33

CC22

GG66

FF33

HH22

EE77

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Earliest Start and Finish TimesEarliest Start and Finish Times

Step 1:Step 1: Make a forward pass through the Make a forward pass through the network as follows: For each activity network as follows: For each activity i i beginning at the Start nodebeginning at the Start node, , compute:compute:

•Earliest Start TimeEarliest Start Time = the maximum of the = the maximum of the earliest finish times of all activities earliest finish times of all activities immediately preceding activity immediately preceding activity ii. (This is 0 . (This is 0 for an activity with no predecessors.)for an activity with no predecessors.)

•Earliest Finish TimeEarliest Finish Time = (Earliest Start Time) = (Earliest Start Time) + (Time to complete activity + (Time to complete activity i i ).).

The project completion time is the maximum The project completion time is the maximum of the Earliest Finish Times at the Finish node.of the Earliest Finish Times at the Finish node.

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Earliest Start and Finish TimesEarliest Start and Finish Times


3 63 6BB33

6 96 9DD33

0 30 3AA33

3 53 5CC22

12 12 1818

GG666 96 9FF

33

5 75 7HH22

5 125 12EE77

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Latest Start and Finish TimesLatest Start and Finish Times

Step 2:Step 2: Make a backwards pass through the Make a backwards pass through the network as follows: Move sequentially network as follows: Move sequentially backwards from the Finish node to the Start backwards from the Finish node to the Start node. At a given node, node. At a given node, jj, consider all activities , consider all activities ending at nodeending at node j j. For each of these activities, . For each of these activities, ii, compute:, compute:

•Latest Finish TimeLatest Finish Time = the minimum of the = the minimum of the latest start times beginning at node latest start times beginning at node jj. (For . (For node node NN, this is the project completion time.), this is the project completion time.)

•Latest Start TimeLatest Start Time = (Latest Finish Time) - = (Latest Finish Time) - (Time to complete activity (Time to complete activity i i ).).

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Latest Start and Finish TimesLatest Start and Finish Times


3 63 6

6 96 9BB33

6 96 9

9 9 1212

DD33

0 30 3

0 30 3AA33

3 53 5

3 53 5CC22

12 12 181812 12 1818

GG666 96 9

15 15 1818

FF33

5 75 7

16 16 1818

HH22

5 125 12

5 125 12EE77

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Determining the Critical PathDetermining the Critical Path

Step 3:Step 3: Calculate the slack time for each Calculate the slack time for each activity by: activity by:

SlackSlack = (Latest Start) - (Earliest = (Latest Start) - (Earliest Start), or Start), or

= (Latest Finish) - (Earliest = (Latest Finish) - (Earliest Finish).Finish).

15 15 Slide Slide


Activity Slack TimeActivity Slack Time

ActivityActivity ESES EFEF LSLS LFLF SlackSlack A 0 3 0 3 0 (crit.)A 0 3 0 3 0 (crit.) B 3 6 6 9 3B 3 6 6 9 3 C 3 5 3 5 0 (crit.)C 3 5 3 5 0 (crit.) D 6 9 9 12 3D 6 9 9 12 3 E 5 12 5 12 0 (crit.)E 5 12 5 12 0 (crit.) F 6 9 15 18 9F 6 9 15 18 9 G 12 18 12 18 0 (crit.)G 12 18 12 18 0 (crit.) H 5 7 16 18 11H 5 7 16 18 11

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•A A critical pathcritical path is a path of activities, from the is a path of activities, from the Start node to the Finish node, with 0 slack Start node to the Finish node, with 0 slack times.times.

•Critical Path: A – C – E – GCritical Path: A – C – E – G

•The The project completion timeproject completion time equals the equals the maximum of the activities’ earliest finish maximum of the activities’ earliest finish times.times.

•Project Completion Time: 18 daysProject Completion Time: 18 days


17 17 Slide Slide


Critical PathCritical Path


3 63 6

6 96 9BB33

6 96 9

9 9 1212

DD33

0 30 3

0 30 3AA33

3 53 5

3 53 5CC22

12 12 181812 12 1818

GG666 96 9

15 15 1818

FF33

5 75 7

16 16 1818

HH22

5 125 12

5 125 12EE77

18 18 Slide Slide

In the In the three-time estimate approachthree-time estimate approach, the time to , the time to complete an activity is assumed to follow a Beta complete an activity is assumed to follow a Beta distribution. distribution.

An activity’s An activity’s mean completion timemean completion time is: is:

tt = ( = (aa + 4 + 4mm + + bb)/6)/6

An activity’s An activity’s completion time variancecompletion time variance is: is:

22 = (( = ((bb--aa)/6))/6)22

•aa = the = the optimisticoptimistic completion time estimate completion time estimate

•bb = the = the pessimisticpessimistic completion time completion time estimateestimate

•mm = the = the most likelymost likely completion time completion time estimateestimate

Uncertain Activity TimesUncertain Activity Times

19 19 Slide Slide

Uncertain Activity TimesUncertain Activity Times

In the three-time estimate approach, the critical In the three-time estimate approach, the critical path is determined as if the mean times for the path is determined as if the mean times for the activities were fixed times. activities were fixed times.

The The overall project completion timeoverall project completion time is assumed is assumed to have a normal distribution with mean equal to to have a normal distribution with mean equal to the sum of the means along the critical path and the sum of the means along the critical path and variance equal to the sum of the variances along variance equal to the sum of the variances along the critical path.the critical path.

20 20 Slide Slide

Example: ABC Associates Example: ABC Associates

Consider the following project:Consider the following project:

Immed. Optimistic Most Likely PessimisticImmed. Optimistic Most Likely Pessimistic

ActivityActivity Predec.Predec. Time (Hr.Time (Hr.) ) Time (Hr.)Time (Hr.) Time (Hr.)Time (Hr.) A A -- 4 -- 4 6 6 8 8 B B -- 1 -- 1 4.5 4.5

5 5 C C A A 3 3 3 3

3 3 D D A 4 5 A 4 5 6 6 E E A 0.5 1 A 0.5 1

1.51.5 F F B,C 3 4 5 B,C 3 4 5 G G B,C B,C 1 1.5 5 1 1.5 5 H H E,F E,F 5 6 7 5 6 7 I I E,F 2 5 8 E,F 2 5 8 J J D,H D,H 2.5 2.75 2.5 2.75

4.5 4.5 K K G,I 3 5 7 G,I 3 5 7

21 21 Slide Slide

Example: ABC AssociatesExample: ABC Associates


E

Start

A

H

D

F

J

I

K

Finish

B

C

G

E

Start

A

H

D

F

J

I

K

Finish

B

C

G

6666

4444

3333

5555

5555

2222

4444

11116666

3333

5555

22 22 Slide Slide


Activity Expected Times and VariancesActivity Expected Times and Variances

tt = ( = (aa + 4 + 4mm + + bb)/6 )/6 22 = (( = ((bb--aa)/6))/6)22

ActivityActivity Expected TimeExpected Time VarianceVariance A A 6 6 4/9 4/9

B B 4 4 4/9 4/9 C C 3 3 0 0 D D 5 5 1/9 1/9 E E 1 1 1/36 1/36 F F 4 4 1/9 1/9 G G 2 2 4/9 4/9 H H 6 6 1/9 1/9 I I 5 5 1 1 J J 3 3 1/9 1/9 K K 5 5 4/9 4/9

23 23 Slide Slide


Earliest/Latest Times and SlackEarliest/Latest Times and Slack

ActivityActivity ESES EF EF LSLS LFLF SlackSlack A A 0 6 0 6 0 * 0 6 0 6 0 *

B B 0 4 5 9 5 0 4 5 9 5 C 6 9 6 9 0 *C 6 9 6 9 0 * D D 6 11 15 20 9 6 11 15 20 9 E E 6 7 12 13 6 6 7 12 13 6 F F 9 13 9 13 0 * 9 13 9 13 0 * G 9 11 16 18 7G 9 11 16 18 7 H H 13 19 14 20 1 13 19 14 20 1 I I 13 18 13 18 0 * 13 18 13 18 0 * J J 19 22 20 23 1 19 22 20 23 1 K K 18 23 18 23 0 * 18 23 18 23 0 *

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•A A critical pathcritical path is a path of activities, from the is a path of activities, from the Start node to the Finish node, with 0 slack Start node to the Finish node, with 0 slack times.times.

•Critical Path: A – C – F – I – KCritical Path: A – C – F – I – K

•The The project completion timeproject completion time equals the equals the maximum of the activities’ earliest finish maximum of the activities’ earliest finish times.times.

•Project Completion Time: 23 hoursProject Completion Time: 23 hours


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Critical Path (A-C-F-I-K)Critical Path (A-C-F-I-K)

E

Start

A

H

D

F

J

I

K

Finish

B

C

G

E

Start

A

H

D

F

J

I

K

Finish

B

C

G

6666

4444

3333

5555

5555

2222

4444

11116666

3333

5555

0 60 60 60 6

9 139 139 139 13

13 1813 1813 1813 18

9 119 1116 1816 18

13 1913 1914 2014 20

19 2219 2220 2320 23

18 2318 2318 2318 23

6 76 712 1312 13

6 96 96 96 9

0 40 45 95 9

6 116 1115 2015 20

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Probability the project will be completed within 24 Probability the project will be completed within 24 hrshrs

22 = = 22AA + + 22

CC + + 22FF + + 22

HH + + 22KK

= 4/9 + 0 + 1/9 + 1 + 4/9 = 4/9 + 0 + 1/9 + 1 + 4/9

= 2= 2

= 1.414= 1.414

zz = (24 - 23)/ = (24 - 23)/(24-23)/1.414 (24-23)/1.414 = .71= .71

From the Standard Normal Distribution From the Standard Normal Distribution table: table:

P(z P(z << .71) = .5 + .2612 = .71) = .5 + .2612 = .7612.7612

27 27 Slide Slide

PERT/CostPERT/Cost

PERT/CostPERT/Cost is a technique for monitoring costs is a technique for monitoring costs during a project. during a project.

Work packagesWork packages (groups of related activities) with (groups of related activities) with estimated budgets and completion times are estimated budgets and completion times are evaluated.evaluated.

A A cost status reportcost status report may be calculated by may be calculated by determining the cost overrun or underrun for each determining the cost overrun or underrun for each work package.work package.

Cost overrun or underrunCost overrun or underrun is calculated by is calculated by subtracting the budgeted cost from the actual cost subtracting the budgeted cost from the actual cost of the work package. of the work package.

For work in progress, overrun or underrun may be For work in progress, overrun or underrun may be determined by subtracting the determined by subtracting the prorated budget cost prorated budget cost from the actual cost to date. from the actual cost to date.

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PERT/CostPERT/Cost

The The overall project cost overrun or underrunoverall project cost overrun or underrun at a at a particular time during a project is determined by particular time during a project is determined by summing the individual cost overruns and summing the individual cost overruns and underruns to date of the work packages.underruns to date of the work packages.

29 29 Slide Slide

Example: How Are We Doing?Example: How Are We Doing?

Consider the following project network:Consider the following project network:

D

3

Start

A

9

I

4

H

5

E

4

B

8Finish

C

10

J

8

F

4

G

3

D

3

Start

A

9

I

4

H

5

E

4

B

8Finish

C

10

J

8

F

4

G

3

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Earliest/Latest TimesEarliest/Latest Times

ActivityActivity ESES EFEF LSLS LFLF SlackSlack A A 0 9 0 9 0 * 0 9 0 9 0 * B B 0 8 5 13 5 0 8 5 13 5 C C 0 10 7 17 7 0 10 7 17 7 D 8 11 22 25 14D 8 11 22 25 14 E E 8 12 13 17 5 8 12 13 17 5 F F 9 13 13 17 4 9 13 13 17 4 G G 9 12 9 12 0 * 9 12 9 12 0 * H H 12 17 12 17 0 * 12 17 12 17 0 * I I 12 16 21 25 9 12 16 21 25 9 J J 17 25 17 25 0 * 17 25 17 25 0 *

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Activity Status (End of Week 11)Activity Status (End of Week 11)

ActivityActivity Actual CostActual Cost % Complete% Complete A A $6,200 100$6,200 100 B B 5,700 100 5,700 100 C C 5,600 90 5,600 90 D D 0 0 0 0 E E 1,000 25 1,000 25 F F 5,000 75 5,000 75 G G 2,000 50 2,000 50 H H 0 0 0 0 I I 0 0 0 0 J J 0 0 0 0

32 32 Slide Slide


Cost Status ReportCost Status Report (Assuming a budgeted cost of $6000 for each activity)(Assuming a budgeted cost of $6000 for each activity)

ActivityActivity Actual CostActual Cost ValueValue DifferenceDifference A A $6,200 (1.00)x6000 = 6000 $6,200 (1.00)x6000 = 6000 $200 $200 B B 5,700 (1.00)x6000 = 6000 5,700 (1.00)x6000 = 6000 - 300 - 300 C 5,600 (.90)x6000 = 5400 200C 5,600 (.90)x6000 = 5400 200 D D 0 0 0 0 0 0 E E 1,000 (.25)x6000 = 1500 1,000 (.25)x6000 = 1500 - 500 - 500 F F 5,000 (.75)x6000 = 4500 5,000 (.75)x6000 = 4500 500 500 G 2,000 (.50)x6000 = 3000 G 2,000 (.50)x6000 = 3000 -1000 -1000 H H 0 0 0 0 0 0 I I 0 0 0 0 0 0 J J 0 0 0 0 0 0 Totals $25,500 $26,400 $- 900Totals $25,500 $26,400 $- 900

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PERT Diagram at End of Week 11PERT Diagram at End of Week 11

11

14

D

3

Start

11

11

A

0

12.5

16.5

I

4

12.5

17.5

H

5

11

14

E

3

11

11

B

0 Finish

11

12

C

1

17.5

25.5

J

8

11

12

F

1

11

12.5

G

1.5

11

14

D

3

Start

11

11

A

0

12.5

16.5

I

4

12.5

17.5

H

5

11

14

E

3

11

11

B

0 Finish

11

12

C

1

17.5

25.5

J

8

11

12

F

1

11

12.5

G

1.5

RemainingRemainingwork (weeks)work (weeks)

Earliest FinishEarliest Finish

Earliest StartEarliest Start

34 34 Slide Slide


Corrective ActionCorrective Action

Note that the project is currently Note that the project is currently experiencing a $900 cost experiencing a $900 cost underunderrun, but the run, but the overall completion time is now 25.5 weeks or overall completion time is now 25.5 weeks or a .5 week delay. Management should consider a .5 week delay. Management should consider using some of the $900 cost savings and apply it using some of the $900 cost savings and apply it to activity G to assist in a more rapid completion to activity G to assist in a more rapid completion of this activity (and hence the entire project).of this activity (and hence the entire project).

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Critical Path MethodCritical Path Method

In the In the Critical Path Method (CPM)Critical Path Method (CPM) approach to approach to project scheduling, it is assumed that the normal project scheduling, it is assumed that the normal time to complete an activity, time to complete an activity, ttj j , which can be , which can be met at a normal cost, met at a normal cost, ccj j , can be crashed to a , can be crashed to a reduced time, reduced time, ttjj’, under maximum crashing for ’, under maximum crashing for an increased cost, an increased cost, ccjj’.’.

Using CPM, activity Using CPM, activity jj's maximum time reduction, 's maximum time reduction, MMj j , may be calculated by: , may be calculated by: MMj j = = ttjj - - ttjj'. It is '. It is assumed that its cost per unit reduction, assumed that its cost per unit reduction, KKj j , is , is linear and can be calculated by: linear and can be calculated by: KKjj = ( = (ccjj' - ' - ccjj)/)/MMjj..

Documents

1 1 Slide Chapter 10 Project Scheduling: PERT/CPM n Project Scheduling with Known Activity Times n Project Scheduling with Uncertain Activity Times n Considering